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arXiv:2506.23345

Trotterization, Operator Scrambling, and Entanglement

Tianfeng Feng, Cao Yue, Qi Zhao

Published Jun 29, 2025• 0 citations
arXivHTMLPDF

Abstract

Operator scrambling, which governs the spread of quantum information in many-body systems, is a central concept in both condensed matter and high-energy physics. Accurately capturing the emergent properties of these systems remains a formidable challenge for classical computation, while quantum simulators have emerged as a powerful tool to address this complexity. In this work, we reveal a fundamental connection between operator scrambling and the reliability of quantum simulations. We show that the Trotter error in simulating operator dynamics is bounded by the degree of operator scrambling, providing the most refined analysis of Trotter errors in operator dynamics so far. Furthermore, we investigate the entanglement properties of the evolved states, revealing that sufficient entanglement can lead to error scaling governed by the normalized Frobenius norms of both the observables of interest and the error operator, thereby enhancing simulation robustness and efficiency compared to previous works. We also show that even in regimes where the system's entanglement remains low, operator-induced entanglement can still emerge and suppress simulation errors. Our results unveil a comprehensive relationship between Trotterization, operator scrambling, and entanglement, offering new perspectives for optimizing quantum simulations.

DOI
10.48550/arxiv.2506.23345

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Operator scrambling, which governs the spread of quantum information in many-body systems, is a central concept in both condensed matter and high-energy physics. Accurately capturing the emergent properties of these systems remains a formidable challenge for classical computation, while quantum simulators have emerged as a powerful tool to address this complexity. In this work, we reveal a fundamental connection between operator scrambling and the reliability of quantum simulations. We show that the Trotter error in simulating operator dynamics is bounded by the degree of operator scrambling, providing the most refined analysis of Trotter errors in operator dynamics so far. Furthermore, we investigate the entanglement properties of the evolved states, revealing that sufficient entanglement can lead to error scaling governed by the normalized Frobenius norms of both the observables of interest and the error operator, thereby enhancing simulation robustness and efficiency compared to previous works. We also show that even in regimes where the system's entanglement remains low, operator-induced entanglement can still emerge and suppress simulation errors. Our results unveil a comprehensive relationship between Trotterization, operator scrambling, and entanglement, offering new perspectives for optimizing quantum simulations.
A physical relationship for the combination of scrambling via spread complexity and entanglement is characterized by the fidelity of quantum unitary dynamics. Time-integrated quantities for the degree of quantum ergodicity capture state spreading from initial to late times under physically plausible operator perturbations. Selecting maximally entangled state as a initial condition evaluates the scrambling of entanglement in this setting. For this reason, we utilize the Rosenzweig-Porter ensembles across different ergodic regimes. Computed integrated spread complexity and associated integrated quantum state fidelity display a monotonic inverse relationship for maximally entangled states from integrability to chaos. Information bounds in non-local high-energy systems and holographic scrambling can be evaluated with this approach.
A bstract Information scrambling refers to the phenomenon in which local quantum information in a many-body system becomes dispersed throughout the entire system under unitary evolution. It has been extensively studied in closed quantum systems, where it is quantified by operator size growth, revealing deep connections between condensed matter physics, high-energy physics, and quantum information. However, when extending the study of operator size dynamics to open quantum systems, two different definitions of operator size distributions emerge. These definitions are based on different treatments of the bath. In this work, we aim to establish a unified picture for operator size dynamics in open quantum systems, using the solvable Brownian SYK models at generic system size. In particular, we provide the conditions under which the signature of scrambling transition — discovered using one particular definition — appears in operator size dynamics under the other definition. Additionally, we extend previous studies by exploring finite-size effects that are not captured by the scramblon theory. Our results provide a refined understanding of operator size dynamics in open quantum systems.
How fast quantum information scrambles such that it becomes inaccessible by local probes turns out to be central to various fields. Motivated by recent works on spin systems with nonlocal interactions, we study information scrambling in different variants of the Ising model. Our work reveals that nonlocal interactions can induce operator dynamics not precisely captured by out-of-time-order correlators (OTOCs). In particular, the operator size exhibits a slowdown in systems with generic power-law interactions despite a highly nonlinear light cone. A recently proposed microscopic model for fast scrambling does not show this slowdown, which uncovers a distinct analogy between a local operator under unitary evolution and the entanglement entropy following a quantum quench. Our work gives new insights on scrambling properties of systems in reach of current quantum simulation platforms and complements results on possibly observing features of quantum gravity in the laboratory.
The complexity of simulating quantum many-body dynamics, or quantum computations, in the Heisenberg picture is governed by the scrambling of initially simple operators into superpositions of exponentially many Pauli strings. The corresponding expansion coefficients define the Pauli spectrum, whose structure controls the performance of classical algorithms based on truncating Pauli expansions. Here we determine the finite-depth Pauli spectrum of random quantum circuits, both in the noiseless case and in the presence of local noise, through its moments, given by the operator stabilizer Rényi entropies. In noiseless circuits, we uncover a hierarchy in the approach to the fully scrambled regime: low moments equilibrate at relatively short depths, while higher moments, which are sensitive to rare, large-amplitude Pauli coefficients, require parametrically larger depths. In noisy circuits, scrambling competes with an effective suppression of operator spreading. Above a critical error per cycle γcN=O(1)γ_c N=\mathcal{O}(1)γc​N=O(1), the operator fails to reach the fully scrambled distribution and remains supported on an atypically sparse subset of Pauli strings. Conversely, below this scale, we rigorously show that classical simulation remains exponentially hard, demonstrating that finite noise does not automatically imply classical simulability. The resulting noise-induced transition in operator complexity therefore delineates the boundary between intrinsically hard quantum dynamics and those that remain classically accessible.
A fundamental challenge in digital quantum simulation (DQS) is the control of an inherent error, which appears when discretizing the time evolution of a quantum many-body system as a sequence of quantum gates, called Trotterization. Here, we show that quantum localization-by constraining the time evolution through quantum interference-strongly bounds these errors for local observables, leading to an error independent of system size and simulation time. DQS is thus intrinsically much more robust than suggested by known error bounds on the global many-body wave function. This robustness is characterized by a sharp threshold as a function of the Trotter step size, which separates a localized region with controllable Trotter errors from a quantum chaotic regime. Our findings show that DQS with comparatively large Trotter steps can retain controlled errors for local observables. It is thus possible to reduce the number of gate operations required to represent the desired time evolution faithfully.
There is great interest in using near-term quantum computers to simulate and study foundational problems in quantum mechanics and quantum information science, such as the scrambling measured by an out-of-time-ordered correlator (OTOC). Here we use an IBM Q processor, quantum error mitigation, and weaved Trotter simulation to study high-resolution operator spreading in a four-spin Ising model as a function of space, time, and integrability. Reaching four spins while retaining high circuit fidelity is made possible by the use of a physically motivated fixed-node variant of the OTOC, allowing scrambling to be estimated without overhead. We find clear signatures of a ballistic operator spreading in a chaotic regime, as well as operator localization in an integrable regime. The techniques developed and demonstrated here open up the possibility of using cloud-based quantum computers to study and visualize scrambling phenomena, as well as quantum information dynamics more generally.